Solve the inequality involving absolute value. Write your final answer in interval notation \[ |-2 x+9| \leq 15 \]

Step 1 ：The absolute value of a number is its distance from zero on the number line. Therefore, the inequality \(|-2x+9| \leq 15\) means that the distance between \(-2x+9\) and 0 is less than or equal to 15. This can be written as two separate inequalities: \(-15 \leq -2x+9 \leq 15\).

Step 2 ：We can solve these inequalities separately to find the solution set for x.

Step 3 ：The solutions to the inequalities are \(x = -3\) and \(x = 12\).

Step 4 ：This means that the solution to the original inequality \(|-2x+9| \leq 15\) is the interval \([-3, 12]\).

Step 5 ：Final Answer: The solution to the inequality is \(\boxed{[-3, 12]}\).